In the Minimal Supersymmetric Standard Model (MSSM) searches for the heaviest CP-even and CP-odd Higgs H, A to tau-lepton pairs severely constrain the parameter region for large values of tanβ and light Higgs bosons H, A. We demonstrate how the exper- imental constraint can be avoided by new decays to light third-generation sfermions, whose left-right couplings to H can be maximised in regions of large trilinear couplings Ab, Aτ for sbottoms and staus, or large supersymmetric (SUSY) Higgs mass μ for stops. Due to the tanβ-enhancement in the production cross-sections via gluon-fusion and in association with bottom-quark pairs for H and A, we find that down-type sfermions, in particular, sbottoms perform a better job in allowing more parameter space than up-type sfermions such as stops, which require much larger values of μ to compensate for tanβ. Vacuum stability as well as flavour observables constraints and direct searches for SUSY particles are imposed. We also associate the lightest CP-even Higgs with the observed 125 GeV SM-like Higgs and impose the experimental constraints from the LHC.